Joint functional calculus in algebra of polynomial tempered distributions
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Functional calculus for generators of operator semigroups, polynomials on locally convex spaces, infinite parameter operator semigroupsAbstract
In this paper we develop a functional calculus for a countable system of generators of contraction strongly continuous semigroups. As a symbol class of such calculus we use the algebra of polynomial tempered distributions. We prove a differential property of constructed calculus and describe its image with the help of the commutant of polynomial shift semigroup. As an application, we consider a function of countable set of second derivative operators.References
N. I. Akhiezer, Lectures on integral transforms, Translations of Mathematical Monographs, vol. 70, American Mathematical Society, Providence, RI, 1988. MathSciNet
Boris Baeumer, Markus Haase, and Mihaly Kovacs, Unbounded functional calculus for bounded groups with applications, J. Evol. Equ. 9 (2009), no. 1, 171-195. MathSciNet CrossRef
Y. M. Berezansky and Y. G. Kondratiev, Spectral methods in infinite-dimensional analysis. Vol. 1, Mathematical Physics and Applied Mathematics, vol. 12/1, Kluwer Academic Publishers, Dordrecht, 1995. MathSciNet CrossRef
Ju. M. Berezans′kii and Ju. S. Samoilenko, Nuclear spaces of functions of an infinite number of variables, Ukrain. Mat. v Z. 25 (1973), 723-737, 860. MathSciNet
Y. M. Berezansky, Z. G. Sheftel, and G. F. Us, Functional analysis. Vol. II, Operator Theory: Advances and Applications, vol. 86, Birkhauser Verlag, Basel, 1996. MathSciNet
H.-J. Borchers, Algebras of unbounded operators in quantum field theory, Phys. A 124 (1984), no. 1-3, 127-144. MathSciNet CrossRef
H.-J. Borchers, On structure of the algebra of field operators, Nuovo Cimento (10) 24 (1962), 214-236. MathSciNet
Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York Inc., New York, 1967. MathSciNet
Sean Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. MathSciNet CrossRef
Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Soc. No. 16 (1955), 140. MathSciNet
Markus Haase, The functional calculus for sectorial operators, Operator Theory: Advances and Applications, vol. 169, Birkhauser Verlag, Basel, 2006. MathSciNet CrossRef
Takeyuki Hida, Hui-Hsiung Kuo, Jurgen Potthoff, and Ludwig Streit, White noise, Mathematics and its Applications, vol. 253, Kluwer Academic Publishers Group, Dordrecht, 1993. MathSciNet CrossRef
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. MathSciNet
Nigel Kalton, Peer Kunstmann, and Lutz Weis, Perturbation and interpolation theorems for the $H^ infty$-calculus with applications to differential operators, Math. Ann. 336 (2006), no. 4, 747-801. MathSciNet CrossRef
N. A. Kachanovsky, Elements of a non-Gaussian analysis on the spaces of functions of infinitely many variables, Ukrainian Math. J. 62 (2011), no. 9, 1420-1448. MathSciNet
H. Komatsu, An Introduction to the Theory of Generalized Functions, University Publ., Tokyo, 2000.
Yuri G. Kondratiev, Ludwig Streit, Werner Westerkamp, and Jia-an Yan, Generalized functions in infinite-dimensional analysis, Hiroshima Math. J. 28 (1998), no. 2, 213-260. MathSciNet
C. Kriegler, Functional calculus and dilation for $C_ 0$-groups of polynomial growth, Semigroup Forum 84 (2012), no. 3, 393-433. MathSciNet CrossRef
O. V. Lopushanskii and S. V. Sharin, A generalized functional calculus of Hille-Phillips type for multiparameter semigroups, Sibirsk. Mat. Zh. 55 (2014), no. 1, 131-146. MathSciNet
Oleh Lopushansky and Sergii Sharyn, Polynomial ultradistributions on $Bbb R^ d_ +$, Topology 48 (2009), no. 2-4, 80-90. MathSciNet CrossRef
A. R. Mirotin, On some properties of the multidimensional Bochner-Phillips functional calculus, Sibirsk. Mat. Zh. 52 (2011), no. 6, 1300-1312. MathSciNet CrossRef
Nobuaki Obata, White noise calculus and Fock space, Lecture Notes in Mathematics, vol. 1577, Springer-Verlag, Berlin, 1994. MathSciNet
Yu. S. Samoilenko, Spectral theory of families of selfadjoint operators, Mathematics and its Applications (Soviet Series), vol. 57, Kluwer Academic Publishers Group, Dordrecht, 1991. MathSciNet CrossRef
Helmut H. Schaefer, Topological vector spaces, Springer-Verlag, New York-Berlin, 1971. MathSciNet
S. V. Sharin, On the cross-correlation operation for Schwartz distributions, Mat. Metodi Fiz.-Mekh. Polya 41 (1998), no. 4, 67-72. MathSciNet CrossRef
A. G. Smirnov, On topological tensor products of functional Frechet and DF spaces, Integral Transforms Spec. Funct. 20 (2009), no. 3-4, 309-318. MathSciNet CrossRef
Armin Uhlmann, Uber die Definition der Quantenfelder nach Wightman und Haag, Wiss. Z. Karl-Marx-Univ. Leipzig Math.-Nat. Reihe 11 (1962), 213-217. MathSciNet
V. S. Vladimirov, Generalized functions in mathematical physics, ``Mir'', Moscow, 1979. MathSciNet
V. V. Zarinov, Compact families of locally convex spaces and FS and DFS spaces, Uspekhi Mat. Nauk 34 (1979), no. 4(208), 97-131, 256. MathSciNet
